When is the chromatic quasisymmetric function symmetric?
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913968842342400 |
|---|---|
| author | Gillespie, Maria Pappe, Joseph Salois, Kyle |
| author_facet | Gillespie, Maria Pappe, Joseph Salois, Kyle |
| contents | We investigate the problem of when a chromatic quasisymmetric function (CQF) $X_G(x;q)$ of a graph $G$ is in fact symmetric. We first prove the remarkable fact that if a product of two quasisymmetric functions $f$ and $g$ in countably infinitely many variables is symmetric, then in fact $f$ and $g$ must be symmetric. This allows the problem to be reduced to the case of connected graphs.
We then show that any labeled graph having more than one source or sink has a nonsymmetric CQF. As a corollary, we find that all trees other than a directed path have a nonsymmetric CQF. We also show that a family of graphs we call ''mixed mountain graphs'' always have symmetric CQF. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_10556 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | When is the chromatic quasisymmetric function symmetric? Gillespie, Maria Pappe, Joseph Salois, Kyle Combinatorics Primary: 05E05 Secondary: 05C15 We investigate the problem of when a chromatic quasisymmetric function (CQF) $X_G(x;q)$ of a graph $G$ is in fact symmetric. We first prove the remarkable fact that if a product of two quasisymmetric functions $f$ and $g$ in countably infinitely many variables is symmetric, then in fact $f$ and $g$ must be symmetric. This allows the problem to be reduced to the case of connected graphs. We then show that any labeled graph having more than one source or sink has a nonsymmetric CQF. As a corollary, we find that all trees other than a directed path have a nonsymmetric CQF. We also show that a family of graphs we call ''mixed mountain graphs'' always have symmetric CQF. |
| title | When is the chromatic quasisymmetric function symmetric? |
| topic | Combinatorics Primary: 05E05 Secondary: 05C15 |
| url | https://arxiv.org/abs/2412.10556 |